To Untangle Multi-Circuit Neural Dynamics, All You Need Is Lag

The brain computes through interactions between neural populations that shape how their activity evolves over time. These interactions often span multiple brain regions, thus forming brain-wide networks that reconfigure as the animal shifts task contexts, behaviors, and decisions. Within these networks, neural signals travel at finite speed, so one population drives another only after a delay whose length reflects both the wiring involved and the animal's current state. Dynamical systems are a natural way to model such lagged interactions, yet existing models often assume a fixed lag, allow several lags but fix the interaction structure in time, or permit richer lag structure at the cost of uninterpretable dynamics. Here, we present Multi-Lag decomposed Linear Dynamical Systems (ML-dLDS), a framework that captures how time-varying neural dynamics arise from multiple co-occurring processes acting across several lags. ML-dLDS represents neural activity as a sparse decomposition of core neural interactions, or circuits, which are reused over time and act across a sparse set of lags. Each circuit-lag pair carries its own time-varying coefficient that sets its momentary contribution to the overall network dynamics, which thereby shift from moment to moment as the active circuit combination changes. We fit the circuits, their per-lag gains, and their time-varying coefficients through an alternating optimization under structured-sparsity penalties. We first validate ML-dLDS on synthetic datasets, where it recovers the ground-truth circuits and their lags, and then demonstrate it on multi-region electrophysiological recordings, where it reveals lag structure that single-lag models oversimplify. Finally, we apply ML-dLDS to COVID-19 case data, where it uncovers interpretable multi-lag structure across countries.